A Question of Boundaries

One of the recurring questions in EROI analysis is where the boundaries should be drawn. How wide do you make the boundaries when calculating energy inputs? Do you include the roads wihch are needed to transport solar panels, for example? Should you include first world salaries of solar panel installers?

The new net energy metric I introduced in the prior article, and the ensuing discussion, can inform where the boundaries should be drawn.. When calculating a rate of exponential growth, the boundaries for net energy analysis should include the minimum energy investment to replicate an energy gathering device. Let me provide some examples.

In our scheme outlined in the prior article, energy investments would not include the entire transportation infrastructure or first world standards of living, which would happen after the exponential growth had already occurred. Energy investments would include, however, the energy investment needed to refine oil, in addition to its extraction, because refined oil products (such as diesel) are required to power the machinery to drill new oil wells.

As another example, refinery losses for oil production would count as an ongoing investment, because the refining doesn’t all need to occur at once before any oil from a well is produced. In this case, it makes no difference if the energy comes from the oil itself, or from some external source, since the effect is the same mathematically in either case. However, self-consumption of coal for underground coal gasification (for example) should simply be ignored and not countaed as an investment at all, since it never needed to be extracted in the first place, so it has no effect on replication time.

As another exmaple, the roads needed for installing solar panels would count as an energy investment, but any other transportation infrastructure would count as a return. The rest of the transportation network is not necessary for replication and could come after the exponential growth has occurred.

In this manner, the disagreements over boundaries could be narrowed considerably.

Of course, these above remarks hold only when calculating a replication time and its effect on economic growth. Another use of net energy metrics is to calculate CO2 emissions. In which case, the boundaries should be drawn differently. When calculating CO2 emissions, self-consumption of cola for underground coal gasification should be counted. Also, first-world transportation (car transportation, for example) to a solar power site should be counted. It depends upon the use to which the metric is being put.

Net energy metrics are only useful as part of a broader calculation, and the broader calculation informs where the boundaries should be drawn.

Net Energy and Economic Growth

One of the effects of net energy metrics is their effect on economic growth. A higher EROI or lower payback time would enable faster exponential growth, other things being equal. This is because a shorter energy payback time would allow more generations of exponential growth in a given span of time.

Any energy gathering device (such as a solar panel or oil well) requires an investment of energy now to obtain more energy later. In turn, the output of that energy gathering device allows us to build more such devices in the future. This re-investment over time, across generations of energy gathering devices, is what allows the growth of energy obtained and is one component of economic growth. Net energy metrics determine how much of our net energy we must sacrifice now to obtain more later, and how fast the subsequent growth can occur.

In this article I will examine the effect of net energy on economic growth. I will introduce a new net energy metric — energy replication time — which can be used to measure the effect of net energy on economic growth. I will also provide some simple mathematical examples, using the new metric, which show the effects of net energy on economic growth. At the end, I’ll describe some of the implications of this new metric.

A New Net Energy Metric

As mentioned before, the amount of net energy we obtain is a function of continued re-investment of energy in the past. This continued re-investment of energy, across generations of energy gathering devices, is part of what constitutes economic growth. In order to calculate the rate of growth, for a given sacrifice of net energy now, we must know the replication time of an energy gathering device. After we have determined the replication time, we can plug that number into standard exponential formulas.

Here we define a new net energy metric, namely Energy Replication Time, or ERT. An ERT can be defined as the upfront energy investment from net ongoing returns. Thus, ERT can be defined as follows: 

U_i / (O_r - O_i)

where U_i is an upfront energy investment, O_r is ongoing power returns, and O_i is ongoing power investments. For example, if we had solar panels with a 1 megawatt-year initial investment, and 1.1 megawatts of power returned, and 0.1 megawatts of ongoing investments, then the energy replication time is 1 year. A panel would return enough net energy in one year to build another panel like it.

It is clear that ongoing investments should simply be subtracted from ongoing returns, as above, when calculating the energy replication time. This is because the ongoing returns are diverted to "pay" the energy investment of a subsequent energy gathering device. For example, assume a hypothetical fusion reactor which fires lasers at deuterium to cause fusion. The lasers require 100MW, and the fusion yields 1,000MW. Of course, you could just run a wire from the output of the reactor back around to the input, thereby powering itself. This leaves 900MW  to "pay" the the energy investment of the next reactor. As a result, the denominator is simply the gross ongoing returns minus ongoing investments.

This new metric (ERT) is slightly different from energy payback time (EPBT). Whereas EPBT considers all energy investment (including disposal) as an upfront investment, our new metric (ERT) considers as an upfront investment only the energy needed to construct a new device and start it operating. As a result, ERT cam be plugged into standard exponential formulas whereas EPBT cannot.

Thus, we have defined a new net metric here, Energy Replication Time (ERT), which is defined as U_i / (O_r - O_i), where U_i is an upfront energy investment, O_r is ongoing power returns, and O_i is ongoing power investments. This metric can be plugged into standard exponential formulas and will determine the amount of growth which can be obtained given a sacrifice of net energy now.

A Few Simple Demonstrations

Using this new metric, we can perform some simple mathematical demonstrations. Assume we have a kind of solar panels with an energy replication time (as defined above) of 1 year. If we diverted all of our energy to building new solar panels, the number of panels would double in one year.

However, that is not exactly right. That would only be correct if the new solar panels were all stored in a dark warehouse until the end of the year, then all suddenly deployed at once. If the solar panels were deployed instantaneously, then the number of solar panels would increase faster than that. A solar panel which is manufactured at the start of the year and deployed right away would start contributing to making new solar panels immediately. The faster the panels are deployed, the faster the growth. 

The exponential base can be calculated using the following formula:

(1+1/n)^n

where n is the replication time divided by the average deployment delay. As a result, if the panels are deployed one month after their manufacture, then the number of panels would increase by a factor of ~2.61 every year (because (1+1/12)^12 = ~2.61). Of course, panels which are deployed instantaneously would increase their number by a factor of e each replication period.

For the rest of this article, I will assume that solar panels are deployed instantaneously. This is a theoretical upper bound of the rate of growth for a given replication period.

Using the formulas outlined above, we can perform some simple calculations of economic growth.

Suppose we have panels with an energy replication time of one year. We wish to increase the number of panels by a factor of 100. If we devote all the energy returned to building new panels, how long would it take to increase the number of panels by 100x? The answer is ln(100), or 4.6 years.

Of course, that estimate is theoretically possible, but wildly unrealistic, for several different reasons. No civilization would devote all of its current energy to investment, for example.

Let’s suppose a civilization diverts 1% of its net energy to building new solar panels continuously, in addition to the energy investment it was already making. In which case, how long would it take to increase the total number of panels by a factor of 100x? The answer is:

ln(100 * (100/1) )

… or 9.21 years, with an initial investment of 1%. However, this estimate is still very unrealistic. While theoretically possible, the vast majority of those new panels would be built in the final year. There would be a sudden onrush of panels in the final year until we obtained 100x as many, as per our goal, and then few would be needed for a long time. That would require building many solar panel factories which operated for less than one year then shut down. It would also require retiring all of our coal fired plants (or whatever we were using previously) in a single year, many of them prematurely. It would also probably require diverting much the workforce to installing solar panels for less than a single year, which would be very disruptive.

Instead, let's try installing new solar panels at a constant higher rate. We'll spread out the new solar panels over a period of 25 years, which is the lifetime of a solar panel, and also of a factory. In which case, the solar panels are phased in gradually.

In which case, we can divide our growth into two phases. First, we have a period of exponential growth, during which an initial investment is grown to an amount sufficient to manufacture additional panels at a new higher rate. Second, the output from those panels could power the manufacture of additional panels at a new, constant, higher rate.

Let’s try another simple mathematical example. Assume a hypothetical civilization which obtains 1GW continuously from solar panels now (drawing from batteries at night). We wish to increase our energy obtained by 100x. In which case, we must install solar panels that return 4GW continuously, each year, for 25 years (4*25=100). Furthermore, we are willing to divert only 1% of our net energy now to initiate the growth. How long of a delay is there before the 1% initial investment can be grown exponentially in order to power the solar panel factories continuously and produce new panels at the higher rate for 25 years? The answer is ln(400), or 5.99 years.

Here we have a more realistic example. We start by building new solar panel factories capable of producing panels that return 4GW continuously per year. We assume that the new factories take 7 years to construct. We then divert 1% of our net energy now to making new panels. During the first phase, the output of all the new panels is continuously re-invested to making more panels, enabling exponential growth. This intermediate phase would last 5.99 years. After which, we can manufacture new panels at a constant higher rate, of 4GW per year, leading to 100x more energy obtained after 25 years. Most of the time, the panels are manufactured and deployed a constant higher rate. 

Thus, if we wish to increase the amount of energy we obtain each year from solar panels by a factor of 100, but do so in a non-disruptive way, it would take a total of 7+5.88+25 years, or 37.88 years. It would take 7 years to build additional solar panel factories (assumed), 5.88 years of exponential growth during which the initial 1% investment is continuously re-invested, then 25 years of manufacturing panels at the higher constant rate. At the end of the 37.88 year period, our hypothetical civilization would obtain 100x as much power continuously as originally, by sacrificing 1% of its energy for 5.88 years. Presumably, we would stagger the building of solar panel factories so they would come online exactly when the panels to power them were installed.

We could also perform a basic calculation of how long it would take to replace our current energy system. Suppose we discover that we have a total of 21 years remaining of all coal, oil, gas, and uranium. We’re willing to invest only 1% of our current net energy for 5 years or less to building a replacement energy system of solar panels. No solar panels have been deployed so far, and no panel factories exist. Will we be able to replace our current energy system entirely before running out of fuel? The answer is yes, but barely. It would take 33.9 years (7+1.39+25), using the assumptions above (the term of 1.39 is how long exponential growth of 1% must continue to produce 100%/25 each year). However, during the final 25 years, the energy from fossil fuels would decline linearly to zero, so fossil fuel usage would average only half during that period, so only 12.5 years of fossil fuel usage during that period, so 20.89 years total (7+1.39+12.5) of fossil fuels would be required to replace our current energy system.

It must also be mentioned that the energy investment for disposal has been omitted here. The above calculations would only be correct if we simply abandoned old solar panels and did nothing to dispose of them. This factor is omitted because it considerably complicates the calculations above. However, it is clear that disposal is an extremely minor factor under any regime of exponential growth, because the disposal would happen 25 years after the initiation of growth (assuming solar panels or other energy gathering devices last 25 years).

Conclusions

From the above, we can draw a few preliminary conclusions.

First, the payback time of solar panels is already short enough to enable extremely rapid exponential growth. We could also undergo extremely rapid exponential growth with any other common source of energy, such as coal, gas, oil, uranium, and so on, because they all have payback times shorter than solar PV.

Second, we are curtailing the exponential growth of energy because of insufficient need. The rest of the economy does not grow anywhere near that fast. We can already grow our energy supply quickly enough for any reasonable anticipated need.

Third, the important energy replication time is for solar panels in a desert near the equator, such as the Sahara or northwest Australia. This is because the panel factories, and the panels to power them, could be located there. As a result, the exponential growth could occur there (over only a few years) and then the resulting panels could be exported elsewhere. The rapid exponential growth would occur regardless of the net energy performance elsewhere, and the export of panels would happen after the exponential growth had occurred. It is mathematically obvious that it does not matter much what the net energy performance is of the panels located elsewhere, as long as it remains above a very low level (like low single digits). Even an EROI of lower than 3 for those panels would be perfectly tolerable when considering the amount of energy generated and how quickly we could grow our energy supply.

Fourth, there is no energy crisis. Even if fossil fuels declined to zero over only a few decades, there would still be enough time to replace them with renewables.

In conclusion. We have introduced a new net energy metric in this article: Energy Replication Time (or ERT). ERT is defined as the upfront investment from net ongoing returns for any energy gathering device (such as solar panels). It was shown that the replication time of solar PV is already short enough to enable rapid exponential growth. As a result, no energy crisis will ever occur. Further improvements in net energy for solar PV (or any other energy source) would have only modest benefits.